Question

triangles abc and dbc have the following characteristics:
\overline{bc} is a side of both triangles
\angle acb and \angle dcb are right angles
\overline{ac} \cong \overline{dc}
which congruence theorem can be used to prove \triangle abc \cong \triangle dbc?
\bigcirc aas
\bigcirc sss
\bigcirc hl
\bigcirc sas

Explanation:

Step1: Identify triangle type

Both $\triangle ABC$ and $\triangle DBC$ are right triangles (right angles at $\angle ACB$ and $\angle DCB$).

Step2: List given congruences

• Shared hypotenuse: $\overline{BC} \cong \overline{BC}$ (reflexive property)
• Leg congruence: $\overline{AC} \cong \overline{DC}$

Step3: Match to congruence theorem

HL (Hypotenuse-Leg) applies to right triangles with congruent hypotenuse and one leg.

Answer:

HL